The Fair Head Start in a Drag Race

Here is an example of one of the racing problems that Prof.
Tapia solved using mathematical arguments.

The National Hot Rod Association (NHRA), the world's largest motorsports
governing body, recently contacted Professor Richard Tapia regarding a
problem they wanted to solve. They were interested in holding a "fair" drag
race between two vehicles of different performance potentials and needed
some expert mathematical help.

Professor Tapia, mathematician and professor in the Department of
Computational and Applied Mathematics at Rice University has had a love of
cars and racing all his life. He and his brother Bobby Tapia as teenagers
built a car that set a world speed record.

An NHRA drag race is an acceleration contest from a standing start between
two vehicles over a measured distance of a quarter-mile (1320 feet). The
drag race was to be held between a dragster, the ultimate acceleration
machine, and a funny car. A funny car is a dragster outfitted with a
carbon-fiber or graphite body that must be a 1991 or later model two-door
coupe or two-door sedan originally mass-produced by an automobile
manufacturer.

Because of air resistance, the graphite shell on the funny car causes it to
be slower than the dragster. The current (February 2000) world record for
the quarter-mile for a dragster is 4.486 seconds set by Larry Dixon. John
Force set the world record for the funny car by completing the quarter-mile
in 4.788 seconds. The difference between the best times for each car makes
it necessary to make some adjustments in the way a race between the two is
designed in order for the race to be competitive.

The NHRA forwarded to Professor Tapia the following data which provides the
time that the funny car reached specific distances during the world record
quarter-mile race.

Time in seconds:

00.885

2.277

3.266

4.091

4.612

4.788

Distance in feet:

60

330

660

1000

1254

1320

The NHRA decided to design a fair race between the dragster and the funny
car where they would give the funny car a head start, so the problem that
they pose to Professor Tapia is the following:

Assume that both cars start at the same time, and the dragster starts at
the beginning of the quarter mile. How far forward should the funny car
start the race so that the race is fair in the sense that both cars would
be expected to cross the quarter mile finish line at the same time?

Professor Tapia examined the data and decided that the best way to answer
this question was to determine where the funny car should be on the
quarter-mile track when the dragster crosses the finish line. The distance
between this point and the finish line is how far forward the funny car
should be at the start of the "fair" race. Can you see why? Essentially,
Professor Tapia shortened the race for the funny car by allowing it to only
cover the portion of the quarter-mile track that it can complete in the
same amount of time required for the dragster to cover the entire track.
With this idea in mind, Professor Tapia was able to discover that the funny
car should begin the race approximately 140 feet ahead of the dragster.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.